Found problems: 821
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$
A lattice point is defined as a point on the plane with integer coordinates. Show that for all positive integers $n$, there is a circle on the plane with exactly n lattice points in its interior (not including its boundary).
Points $A$ and $B$ are chosen on a circle $k$. Let AP and $BQ$ be segments of the same length tangent to $k$, drawn on different sides of line $AB$. Prove that the line $AB$ bisects the segment $PQ$.
Two circles $S_1$ and $S_2$, of radii $6$ units and $3$ units respectively, are tangent to each other, externally. Let $AC$ and $BD$ be their direct common tangents with $A$ and $B$ on $S_1$, and $C$ and $D$ on $S_2$. Find the area of quadrilateral $ABDC$ to the nearest Integer.
For any integer $n\geq1$, we consider a set $P_{2n}$ of $2n$ points placed equidistantly on a circle. A [i]perfect matching[/i] on this point set is comprised of $n$ (straight-line) segments whose endpoints constitute $P_{2n}$. Let $\mathcal{M}_{n}$ denote the set of all non-crossing perfect matchings on $P_{2n}$. A perfect matching $M\in \mathcal{M}_{n}$ is said to be [i]centrally symmetric[/i], if it is invariant under point reflection at the circle center. Determine, as a function of $n$, the number of centrally symmetric perfect matchings within $\mathcal{M}_{n}$.
[i]Proposed by Mirko Petrusevski[/i]
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$?
$\textbf{(A) } 0\qquad \textbf{(B) } \sqrt{\frac{2}{3}}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\sqrt{\frac{3}{2}}$
Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square?
(C.Pohoata, A.Zaslavsky)
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
A triangle $ABC$ is inscribed in the circle $(O,R)$. A circle $(O',R')$ is internally tangent to $(O)$ at $I$ such that $R < R'$. $P$ is a point on the circle $(O)$. Rays $PA, PB, PC$ meet $(O')$ at $A_1,B_1,C_1$. Let $A_2B_2C_2$ be the triangle formed by the intersections of the line symmetric to $B_1C_1$ about $BC$, the line symmetric to $C_1A_1$ about $CA$ and the line symmetric to $A_1B_1$ about $AB$. Prove that the circumcircle of $A_2B_2C_2$ is tangent to $(O)$.
Nguyễn Văn Linh
A circle with center $O$ passes through points $A$ and $C$ and intersects the sides $AB$ and $BC$ of triangle $ABC$ at points $K$ and $N$, respectively. The circumcircles of triangles $ABC$ and $KBN$ meet at distinct points $B$ and $M$. Prove that $\angle OMB = 90^o$.
Circles $o_1$ and $o_2$ with centers in $O_1$ and $O_2$, respectively, intersect in two different points $A$ and $B$, wherein angle $O_1AO_2$ is obtuse. Line $O_1B$ intersects circle $o_2$ in point $C \neq B$. Line $O_2B$ intersects circle $o_1$ in point $D \neq B$. Show that point $B$ is incenter of triangle $ACD$.
Let $k_1$ and $k_2$ be two circles with centers $O_1$ and $O_2$ and equal radius $r$ such that $O_1O_2 = r$. Let $A$ and $B$ be two points lying on the circle $k_1$ and being symmetric to each other with respect to the line $O_1O_2$. Let $P$ be an arbitrary point on $k_2$. Prove that
\[PA^2 + PB^2 \geq 2r^2.\]
Two circles $(U)$ and $(V)$ intersect at $A,B$. A line d meets $(U), (V)$ at $P, Q$ and $R,S$ respectively. Let $t_P, t_Q, t_R,t_S$ be the tangents at $P,Q,R, S$ of the two circles. Another circle $(W)$ passes through through $A, B$. Prove that if the circumcircle of triangle that is formed by the intersections of $t_P,t_R, AB$ is tangent to $(W)$ then the circumcircle of triangle formed by $t_Q, t_S, AB$ is also tangent to $(W)$.
Tran Minh Ngoc, a student of Ho Chi Minh City College, Ho Chi Minh
Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too.
Let $ABC$ be a triangle. Consider three circles, centered at $A, B, C$, with respective radii $$\sqrt{AB \cdot AC},\sqrt{BC \cdot BA},\sqrt{CA \cdot CB}.$$ Given that there are six distinct pairwise intersections between these three circles, show that they lie on two concentric circles.
[i](Two circles are concentric if they have the same center.)[/i]
Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.
A circle $C$ with center $O$ on base $BC$ of an isosceles triangle $ABC$ is tangent to the equal sides $AB,AC$. If point $P$ on $AB$ and point $Q$ on $AC$ are selected such that $PB \times CQ = (\frac{BC}{2})^2$, prove that line segment $PQ$ is tangent to circle $C$, and prove the converse.
A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$
Let $C_1(O_1)$ and $ C_2(O_2)$ be two circles such that $C_1$ passes through $O_2$. Point $M$ lies on $C_1$ such that $M \notin O_1O_2$. The tangents from $M$ at $O_2$ meet again $C_1$ at $A$ and $B$. Prove that the tangents from $A$ and $B$ at $C_2$ - others than $MA$ and $MB$ - meet at a point located on $C_1$.
Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.
Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$.
Prove that the lines $AB$ and $PQ$ are perpendicular.
Three circles of radius $1$ are inscribed in a square of side length $s$, such that the circles do not overlap or coincide with each other. What is the minimum $s$ where such a configuration is possible?
Two circles $k_1(O_1,R)$ and $k_2(O_2,r)$ are given in the plane such that $R \ge \sqrt2 r$ and $$O_1O_2 =\sqrt{R^2 +r^2 - r\sqrt{4R^2 +r^2}}.$$ Let $A$ be an arbitrary point on $k_1$. The tangents from $A$ to $k_2$ touch $k_2$ at $B$ and $C$ and intersect $k_1$ again at $D$ and $E$, respectively. Prove that $BD \cdot CE = r^2$
Prove that the circle with radius $2$ can be completely covered with $7$ unit circles